In the context of importance sampling ratio, how is the equation $\mathbb{E}\left[\rho_{t: T-1} G_{t} | S_{t}=s\right]=v_{\pi}(s)$ derived?
It has nothing to do with Bayes rule, like you said, $G_t$ is the return we get by following the policy $b$ , so the value of the state we get that return in is equal to the expected value of that return. On page 59 of the book, equation 3.14 derives the value of the state as expected value of the return but in this case we are following the behaviour policy $b$ so you would replace $\pi (a \mid s)$ with $b (a \mid s)$ and $v_\pi (s')$ with $v_b (s')$ and you would get the value of the state following the policy $b$, that is $v_b (s)$.
We are actually interested in the value of the state when we follow target policy $\pi$ , but we got the value by following behaviour policy, so we will multiply that expression with importance sampling ratio. If we do that $b (a \mid s)$ terms will cancel out and you are only left with term $\pi (a \mid s)$ from importance sampling ratio so we will get the value by following the target policy $\pi$ that is $v_\pi (s)$